Modeling the kinetics of the neutralizing antibody response against SARS-CoV-2 variants after several administrations of Bnt162b2

Because SARS-CoV-2 constantly mutates to escape from the immune response, there is a reduction of neutralizing capacity of antibodies initially targeting the historical strain against emerging Variants of Concern (VoC)s. That is why the measure of the protection conferred by vaccination cannot solely rely on the antibody levels, but also requires to measure their neutralization capacity. Here we used a mathematical model to follow the humoral response in 26 individuals that received up to three vaccination doses of Bnt162b2 vaccine, and for whom both anti-S IgG and neutralization capacity was measured longitudinally against all main VoCs. Our model could identify two independent mechanisms that led to a marked increase in measured humoral response over the successive vaccination doses. In addition to the already known increase in IgG levels after each dose, we identified that the neutralization capacity was significantly increased after the third vaccine administration against all VoCs, despite large inter-individual variability. Consequently, the model projects that the mean duration of detectable neutralizing capacity against non-Omicron VoC is between 348 days (Beta variant, 95% Prediction Intervals PI [307; 389]) and 587 days (Alpha variant, 95% PI [537; 636]). Despite the low neutralization levels after three doses, the mean duration of detectable neutralizing capacity against Omicron variants varies between 173 days (BA.5 variant, 95% PI [142; 200]) and 256 days (BA.1 variant, 95% PI [227; 286]). Our model shows the benefit of incorporating the neutralization capacity in the follow-up of patients to better inform on their level of protection against the different SARS-CoV-2 variants. Trial registration: This clinical trial is registered with ClinicalTrials.gov, Trial IDs NCT04750720 and NCT05315583.

giving necessary conditions for ODE structural identifiability, we derive that every 11 ODEs: 12 S a = a µV 0 M 1 G(t) − δ S S ȧ Ab = θ a S a − δ Ab Ab ruling the behavior of S a := a S would predict the same evolution for Ab, no matter the 13 value of a. To remove this problem, we consider the ODE : corresponding to a = µV 0 M 1 −1 and ruling the rescaled variable S := µV 0 M 1 −1 S in 15 which the parameter ϑ := µV 0 M 1 θ merges the previous non-identifiable parameters into 16 one, thus re-establishing identifiability.

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Regarding ϑ interpretation, let us remind that first injection is assumed to be the 18 first antigen encounter for vaccinated subjects, so S(t 1 ) = Ab(t 1 ) = 0. From this, let us 19 notice that 20 Ab(t 1 ) = ϑṠ(t 1 ) − δ AbȦ b(t 1 ) = ϑ (G(t 1 ) − δ S S(t 1 )) − δ Ab (ϑS(t 1 ) − δ Ab Ab(t 1 )) = ϑ, 21 so ϑ is the initial acceleration in antibody production after first antigen encounter. Even without this approximation, the original ODE have the following structurally with M = ρ −1 1 M and S = ρ −1 1 S if we assume the generation rate ρ is piecewise constant 29 and given by ρ = ρ 1 f ρ k between [t k , t k+1 ] . At the contrary to the simplified model (3), 30 the ODE (4) still assumes that M undergoes a transient phase which may slow down Ab 31 production right after each injection. To further explore the effect of delay between 32 antigen injection and the start of antibody production, and its consistency with our 33 available data, we also consider the model: represents the latency, quantified by the parameter α, between the generation of the 37 memory compartment and its ability to differentiate into secreting cells. (f ρ2 , f ρ3 , µ S , α) . As in the main paper, we profile on (δ V , δ S ). Not only we face 46 difficulty to estimate µ S but the simplest ODE (3) gives us the lowest AIC=16.5, 47 followed by ODE (5) with AIC=17.6 and finally AIC=18.3 for ODE (4).

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To illustrate why such delay does not seem relevant on a dynamical point of view to generate Ab trajectory. We plot in Figure 1-right the 60 predicted Ab trajectories for µ S ranging from 20 to 200 and the only corresponding 61 prediction from ODE (3). For any µ S value, M quickly settles to its steady-state and 62 the corresponding Ab prediction does not differs from the ones predicted by the simpler 63 ODE (3), supporting its role as relevant practically identifiable approximation of (4).
Before settling to the chosen model for neutralization, different expressions for F (ν, t) 66 have been tested corresponding to different hypotheses on neutralization kinetic. Our 67 goal is to unveil key factors acting on the dynamic of observed gain in neutralization 68 with respect to time. For this, we compare two hypotheses: 69 H1 no gain from sequential injection, the main factor is the elapsed time since first 70 antigen encounters,

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H injection-dependent gain, the main factor is the number of exposition to antigen.

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Under H more precise scenario can be investigated. In particular, does some VoCs 73 benefit more from this repeated injections than others? After the first dose there are 74 differences between neutralization level between VoCs, but additional injections can 75 increase Memory B-cells repertoire diversity [3,4]. Thus, for VoCs with major mutations 76 comparing to the strain toward which the vaccine was primarily directed, an higher  The function F (ν, t) will change according to the tested hypothesis: For H1, γf ν = lim t −→+∞ F (ν, t) represents the maximal neutralization capacity per 86 antibody concentration unit for each VoCs. Parameter β ν is the required time to move 87 from 0 to γfν 2 . It quantifies the speediness of gain, the smaller β ν the quicker the gain 88 will be. H2 is a simplified version of H3 corresponding to g 2,ν := 1 and g 3,ν := 1 that is 89 a constant relative gain for all VoCs.

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Before estimation, we made simplifying assumptions on some parameter values for 91 practical identifiability purposes. For H1, we assume β ν is common to all variants i.e. 92 β ν := β. For H3, we were unable to estimate g 2,ν because most of neutralization data 93 for Omicron were left-censored before third injection, so we restrict testing to g 2,ν := 1 94 thus retrieving the retained model in the main paper with g 3,ν := g ν . We then proceed 95 to parameter estimation for each neutralization model corresponding to hypotheses H1-96 H2-H3 similarly as described in Section Methods with the same underlying 97 mechanistic model (3) for Ab evolution. These estimation leads to significantly different 98 Akaike Information criteria (AIC) values. For each H1, we end up with AIC=1786, for 99 H2 AIC=1747 and for H3 AIC=1627. This drives us to choose H3 over H1-H2 that 100 is the repeated injections as the main driver of affinity gain.

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Finally, our assumption about the linear relationship between ED ν 50 and Ab can be 102 seen as too simple. In particular, it may miss saturating effect when antibody 103 concentration is important and competition for free viruses can emerge. To account for 104 this, we test the sigmoïd model 105 H4 ED ν 50 = F (ν, t) Ab n Ab n +θ n ν with F (ν, t) retained from H3.

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Its estimation leads to AIC=1691 that is, higher than H3. Appendix C: internal model assessment To test the external validity of our model, we assess its capacity to make consistent 127 predictions for a cohort presented in [5][6][7] where, similarly as in our case, antibody 128 concentration are measured in BAU/ml. This cohort originally composed of N=151 129 participants is divided into two subgroups differing by their age, one of N=89 healthcare 130 workers (HCW)s and a second one of N=62 older peoples (OP)s. Their received up to 131 three doses of mRNA vaccines (either BNT162b2 or mRNA-1273), despite differences in 132 the injection schedules between HCWs and OPs, they both received their second 133 injection later than in our analyzed cohort. Of note, measurements were not available 134 for the same subjects after second and third dose, a summary of measured patient 135 characteristics is given in Table 1.

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Antibody concentration are then measured at different timepoints, it is important to 137 note that they are restricted to HCWs prior to the third dose and given for the whole 138 cohort afterward. Regarding predictions, we choose the presented estimated parameter 139 values and we change the injection timing accordingly to this new cohort. Prior to third 140 injection, we took the median injection time among HCWs (i.e. 97 days). For predicting 141 the third dose effect, we took the averaged median injection times for HCWs and OPs 142 weighted by the number of subjects among these subgroups. Measurement and

Measurements
Model predictions Initial estimation f M3 := 3.5f M3 One month after first dose  Table 2. Comparison between measurements and model predictions (both given in log 10 (BAU/ml). The first column presents the raw measurements. The second column presents the predictions made from the estimations presented in the main article. The third column presents the predictions made with the updated parameter f M3 .
predictions are presented in Figure 3 and Table 2.  While the model correctly predicts the antibody level post first and second dose, it 145 slightly under-estimates antibody level after the third dose. This could be simply due to 146 random fluctuations and measurement error but could also reveal genuine 147 immunological mechanisms that are not accounted for in the model. In particular, 148 nearly half of the subjects in the Canadian cohort received different vaccines, which may 149 induce greater immunogenicity [8]. It is also possible that the timing of administration 150 results in a different maturation kinetics Additionally, the injection timing itself can 151 have an impact on the antibody concentration [9] Still, no matter the underlying cause 152 of the observed differences, updating only one parameter, here f M3 ruling the antibody 153 peak after the third dose is enough to re-establish consistency with the last three 154 measurements. This is illustrated in Figure 3 representing the antibody concentration Our proposed model can be easily modified to account for the effect of additional doses, in particular a fourth one, and predict the duration of acquired neutralization. This requires to extend our neutralization model now given by: F (ν, t) = γf ν (1 t<t2 + f 2 1 t∈[t2;t3] + f 3 g ν 1 t∈[t3;t4] + f 4 g 4,ν 1 t≥t4 ) where t 4 is the time of fourth injection, chosen here to be 1 year after t 3 . This model 159 definition requires new parameters, f 4 the fold-change for neutralization gain brought by 160 a fourth injection for D614G and g 4,ν the relative VoC-specific gain compared to D614G. 161 In addition to neutralization parameters, we also have to add f M4 = M4 M1 , the fold-change 162 for the memory compartment at t 4 to account for change in Ab dynamics in ODE (3). 163